(Schur's Triangularization Lemma)
(a) Let U be a subspace of V and fix bases BU ⊆ BV. What is the relationship between the representation of a vector from U with respect to BU and the representation of that vector (viewed as a member of V) with respect to BV?
(b) What about maps?
(c) Fix a basis B = (1, . . . ,n) for V and observe that the spans
[∅] = {} ⊂ [{1}] ⊂ [{1,2 }] ⊂ . . . ⊂ [B] = V
form a strictly increasing chain of subspaces. Show that for any linear map
h: V → W there is a chain W0 = {} ⊆ W1 ⊆ . . . ⊆ Wm = W of subspaces of W such that
h([{1, . . . , i}]) ⊂ Wi
(d) Conclude that for every linear map h: V → W there are bases B,D so the matrix representing h with respect to B,D is upper-triangular (that is, each entry hi,j with i > j is zero).
(e) Is an upper-triangular representation unique?